Towards a homotopy theory of higher dimensional transition systems

Philippe Gaucher

We proved in a previous work that Cattani-Sassone's higher
dimensional transition systems can be interpreted as a
small-orthogonality class of a topological locally finitely
presentable category of weak higher dimensional transition
systems. In this paper, we turn our attention to the full
subcategory of weak higher dimensional transition systems which are
unions of cubes. It is proved that there exists a left proper
combinatorial model structure such that two objects are weakly
equivalent if and only if they have the same cubes after
simplification of the labelling. This model structure is obtained by
Bousfield localizing a model structure which is left determined with
respect to a class of maps which is not the class of monomorphisms.
We prove that the higher dimensional transition systems
corresponding to two process algebras are weakly equivalent if and
only if they are isomorphic. We also construct a second Bousfield
localization in which two bisimilar cubical transition systems are
weakly equivalent. The appendix contains a technical lemma about
smallness of weak factorization systems in coreflective
subcategories which can be of independent interest. This paper is a
first step towards a homotopical interpretation of bisimulation for
higher dimensional transition systems.